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Downloaded from https://academic.oup.com/mts/article-abstract/21/2/200/1084715 by SMT Member Access user on 29 August 2019 Defining Modular Transformations Matthew Santa I. BARTOICS TRANSFORMATION AND ITS GENERALIZATION Examples 1-3 show instances of this transformation in BartOk's music. Example 1, from Music for Strings, Percussion, and Ce- In a lecture at Harvard University in 1943, Bela BartOk ac- lesta, shows how the beginning of the chromatic theme of the first knowledged his discovery and use of a transformation that maps movement, <A B6 C# C B>, maps onto the beginning of the dia- musical entities from chromatic to diatonic collections: tonic (Lydian) theme in m. 204 of the fourth movement, <C D G Working [with] chromatic degrees gave me [an] idea which led to the use E>. That is, scale-degree 1 from the chromatic collection maps of a new device. This consists of the change of the chromatic degrees into onto scale-degree 1 from the diatonic, scale-degree 2 maps onto diatonic degrees. In other words, the succession of chromatic degrees is scale-degree 2, and so forth, as shown by the scale-degree num- extended by leveling them over a diatonic terrain. bers beneath the example. Examples 2 and 3 are taken from You know very well the extension of themes in their values called aug- Mikrokosmos. Example 2 shows how piece No. 64a maps onto mentation, and their compression in value called diminution. These de- No. 64b by a transformation that compresses the diatonic music vices are very well known. Now, this new device could be called into a chromatic space, while Example 3 shows how the theme "extension in range" of a theme. For the extension we have the liberty to from the beginning of piece No. 112 maps onto the theme at the choose any diatonic scale or mode.' beginning of its second section (marked un poco meno mosso) by a transformation that compresses the diatonic music into a chro- matic space. Numerals in parentheses below the staves in Ex- Early versions of this paper were presented at the annual meeting of the ample 3 represent scale degrees that do not participate.' New England Conference of Music Theorists, Storrs, Connecticut, 28 March 1998, and at the annual meeting of the Society for Music Theory, Chapel Hill, But BartOk's transformation need not be limited to mappings North Carolina, 4 December 1998. I wish to thank David Lewin, David Smyth, between diatonic and chromatic spaces; one can find examples of and Joseph Straus for their invaluable comments on earlier drafts. mappings to and from other modular spaces in BartOk's own 'Bela Baru*, "Harvard Lectures," in Bela BartOk Essays, ed. Benjamin music. Example 4 is taken from the first movement of his Fourth Suchoff (New York: St. Martin's Press, 1976), 381. Wayne Alpern has also ex- String Quartet, and shows how the chromatic first theme in m. 7, plored this remodularization technique in BartOk's music from a transforma- tional perspective in "BartOk's Compositional Process: 'Extension in Range' as <B C D6 C B I36>, maps onto the octatonic segment in the first vi- a Progressive Contour Transformation," paper presented at the annual confer- olin part of m. 158, <E6 F G6 F E6 D>. This article formalizes ence of Music Theory Midwest, 15 May 1998. Examples 1, 2, 3, and 12 are taken from Alpern's paper. Alpern's work generalizes this operation to non- , The terms modular spare and modular system will be defined here as syn- modularized progressive contour transformations in Bart6k as well as Berg, onyms referring to specified partitionings of the octave within the larger context whereas this article retains an exclusively modular perspective. of an equal-tempered harmonic system. Defining Modular Transformations 201 Downloaded from https://academic.oup.com/mts/article-abstract/21/2/200/1084715 by SMT Member Access user on 29 August 2019 Example 1. BartOk, Music for Strings, Percussion, and Celesta, I and IV; transformation from chromatic to diatonic 1st mvt., m. 1 4th mvt., m. 204 6- $ 4 <1 2 5 4 3> <1 2 5 4 3> chromatic scale degrees diatonic (Lydian) scale degrees Example 2. BartOk, No. 64a and 64b from Mikrokosmos, mm. 1-4 of each; transformation from diatonic to chromatic 64a. 64b. a pa J 0 O <1 2 3 4 5 4 3 4 1> <1 2 3 4 5 4 3 4 1> diatonic scale steps chromatic scale steps Example 3. BartOk, No. 112 from Mikrokosmos; transformation from diatonic to chromatic mm. 1-8 a J J J 1 17 <1 2 3 4 (4 4) 4 3 2 3 (3 3) 3 2 1 2 (2 2) 2 3 2 1 1> diatonic scale steps mm. 32-39 <1 2 3 4 (7) 4 3 2 3 (6) 3 2 1 2 (5) 2 3 2 1> chromatic scale steps 202 Music Theory Spectrum Downloaded from https://academic.oup.com/mts/article-abstract/21/2/200/1084715 by SMT Member Access user on 29 August 2019 Example 4. BartOk, String Quartet No. 4, I; transformation from chro- the term "scale degree," which has potentially misleading diatonic matic to octatonic and tonal connotations, with the term "step class."4 A step class is defined here as a numbered position within a modular system; oc- m. 7 m. 158 tave equivalence is assumed. Step classes are numbered 0 to n (n being equal to the cardinality of the modulus minus 1). 5 Thus each step class in a mod7 (diatonic) space, for example, will be equal P • to the corresponding scale-degree minus 1 (e.g., scale-degree 4 is <2 3 4 3 2 1> <2 3 4 3 2 1> equivalent to step-class 3). chromatic scale degrees octatonic scale degrees Let us define MODTRANS (x, y, z) as a transformation that maps each step class of a musical entity in modular system x onto Bartok's transformation, and generalizes it to map musical entities a corresponding step class in modular system y, where z represents to and from any one of five different modular spaces: chromatic, the "point of synchronization," the pitch class in the starting mod- octatonic, diatonic, whole-tone, and pentatonic. 3 ulus that is interpreted as step-class 0. From this pointof synchro- It is possible to discuss BartOles transformation in a more sys- nization, one may then construct a table of mappings from the tematic way. Before doing so, however, it is necessary to replace starting modulus to the destination modulus and use the table to map pitch classes from the first musical entity to the second one. 'I have restricted my focus to these spaces because these are the most im- Example 5 displays all of the possible rotations for the mod12, portant subdivisions of the octave into collections of twelve, eight, seven, six, mod8, mod7, mod6, and mod5 systems considered here in integer and five notes, respectively. Though many other seven-note collections could also be considered, I am understanding those to be derivations from a diatonic notation, arranged so that one may easily find their corresponding norm. I exclude harmonic and melodic minor, for example, as derivations from step classes; the table can thus be used to find the mappings for the natural form. It will be easy for the reader to adapt my methods to such de- any MODTRANS operation occurring between the five systems. rived spaces, but it would be overly cumbersome to include them all here. The integer notation in Example 5 sets the point of synchroniza- There is a rich literature on the properties of modular systems. See, for ex- tion equal to 0, and the remaining pitch classes in each modular ample, Milton Babbitt, "The Structure and Function of Music Theory I," system receive a value equal to their distance in semitones above College Music Symposium 5 (1965): 49-60; Hubert Howe, "Some Combina- tional Properties of Pitch Structures," Perspectives of New Music 4/1 (1965): the point of synchronization. 45-61; Carlton Gamer, "Some Combinational Resources of Equal-Tempered Because mappings among systems partitioned into unequal Systems," Journal of Music Theory 11/1 (1967): 32-59; Robert Cogan and steps are variable depending on where in the system the point of Pozzi Escot, Sonic Design (Englewood Cliffs, N.J.: Prentice Hall, 1976); Jay Rahn, "Some Recurrent Features of Scales," In Theory Only 2/11-12 (1977): 43-52; Richmond Browne, "Tonal Implications of the Diatonic Set," In Theory 4Stephen Dembski used the term "step class" in "Steps and Skips from Only 5/6-7 (1981): 3-21; Robert Gauldin, "The Cycle-7 Complex: Relations of Content and Order: Aspects of a Generalized Step-Class System," paper pre- Diatonic Set Theory to the Evolution of Ancient Tonal Systems," Music Theory sented at the annual meeting of the Society for Music Theory, Baltimore, 5 Spectrum 5 (1983): 39-55; John Clough and Gerald Myerson, "Variety and November 1988. Multiplicity in Diatonic Systems," Journal of Music Theory 29/2 (1985): , Because "step class" is defined here as an order position within a modular 249-70; Norman Carey and David Clampitt, "Aspects of Well-Formed Scales," system, there is a danger that it might be confused with the term "order posi- Music Theory Spectrum 11/2 (1989): 187-206; and Jay Rahn, "Coordination of tion." This article advocates using "order position" to refer to ordering in a con- Interval Sizes in Seven-Tone Collections," Journal of Music Theory 35/1 text that is specific to a musical work, such as a twelve-tone row, and reserving (1991): 33-60.
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